Water scarcity is a major challenge in the agriculture industry, and traditional irrigation methods are often wasteful and inefficient. To address this challenge, a smart solar irrigation system that uses loT and Artificial Neural Network (ANN) algorithms can optimize water usage for agriculture. The system can provide automated irrigation, improve crop yields, and reduce water consumption. This paper proposes a design and implementation methodology of a smart solar irrigation system using loT and ANN algorithms. The system includes solar panels, a water pump, a water storage tank, sensors, loT devices, and ANN algorithms. The system is designed to automate the irrigation process by controlling the water pump based on the data collected from the sensors.

Developing number sequence based on polygonal numbers is an enthusiastic field in number theory. As tetrahedral numbers are similar to pyramids, one of the Seven wonders of the World, yields a unique copiousness in its suitability. In number theory study of pyramidal numbers vary in richness and variety. Also the study of continued fractions is a fastly developing field.

Both Farey sequence and continued fractions are recently developing fields of number theory. Their inter relationships are studied in this paper. Farey sequence is analyzed through continued fractions as Farey sequence consist of fractals from 0 to 1. It is considered as sum of rational numbers. An attempt has been made to write the mediant inserted in Farey sequence of order as a matrix whose diagonal elements are unity. For the matrix representation only the new inserted elements have been considered.

In number theory study of polygonal numbers vary in richness and variety. Also the study of continued fractions is a fast developing field. Here in this study an attempt has been made to represent ratios of polygonal numbers with triangular number, square number, pentagonal number, hexagonal number as basis.Notations:1. < p(0) , p(1), p(2) , p(3), ...., p(n)>- continued fraction expansion2. p(3, n)- triangular number3. p(4, n)- square number4. p(5, n)- pentagonal number5. p (6, n)- hexagonal number6. p(m, n)- polygonal number of order and rank 'm' and rang 'n'

Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.

In this paper we present some identities for the sums of squares of Fibonacci and Lucas numbers with consecutive primes, using maximal prime gap ( ) G x x2 ~log , as indices.